A Mathematical Framework for Modeling Axon Guidance
Abstract
In this paper, a simulation tool for modeling axon guidance is presented. A mathematical framework in which a wide range of models can been implemented has been developed together with efficient numerical algorithms. In our framework, models can be defined that consist of concentration fields of guidance molecules in combination with finitedimensional state vectors. These vectors can characterize migrating growth cones, target neurons that release guidance molecules, or other cells that act as sources of membranebound or diffusible guidance molecules. The underlying mathematical framework is presented as well as the numerical methods to solve them. The potential applications of our simulation tool are illustrated with a number of examples, including a model of topographic mapping.
Keywords
Axon guidance Growth cone Topographic mapping Simulation Numerical methods1. Introduction
The proper functioning of the nervous system relies on the formation of correct neuronal connections. During development, neurons project long, thin extensions, called axons, which grow out, often over long distances, to form synaptic connections with appropriate target cells. Axons can find their target cells with remarkable precision by using molecular cues in the extracellular space (for reviews, see TessierLavigne and Goodman, 1996; Dickson, 2002; Yamamoto et al., 2003). They steer axons by regulating cytoskeletal dynamics in the growth cone (Huber et al., 2003), a highly motile and sensitive structure at the tip of a growing axon. Extracellular cues can either attract or repel growth cones, and can either be relatively fixed or diffuse freely through the extracellular space. Target cells secrete diffusible attractants and create a gradient of increasing concentration, which the growth cone can sense and follow (Goodhill, 1997). Cells that the axons have to avoid or grow away from produce repellents. By integrating different molecular cues in their environment, growth cones guide axons along the appropriate pathways and via intermediate targets to their final destination, where they stop growing and form axonal arbors to establish synaptic connections. The responsiveness of growth cones to guidance cues is not static but can change dynamically during navigation. Growth cones can undergo consecutive phases of desensitization and resensitization (Ming et al., 2002), and can respond to the same cue in different ways at different points along their journey (Shirasaki et al., 1998; Zou et al., 2000; Shewan et al., 2002). Through modulation of the internal state of the growth cone, attraction can be converted to repulsion and vice versa (Song et al., 1998; Song and Poo, 1999).
Axon guidance is a very active field of research. Several families of molecules have been identified and a few general mechanisms can account for many guidance phenomena. The major challenge is now to understand, not only qualitatively but also quantitatively, how these molecules and mechanisms act in concert to generate the complex patterns of neuronal connections that are found in the nervous system.
To address this challenge, experimental work needs to be complemented by modeling studies. However, unlike for the study of electrical activity in neurons and neuronal networks (e.g., NEURON; Hines and Carnevale, 1997), there are currently no general simulation tools available for axon guidance.
In Hentschel and van Ooyen (1999) a model is presented in which growing axons on a plain are modeled by means of differential equations for the locations of the growth cones. These equations are coupled to diffusion equations that describe the concentration fields of diffusible chemoattractants and chemorepellents (henceforth referred to as guidance molecules). The system is simplified by using quasisteadystate approximations for the concentration fields. This approach turns the problem of solving a system consisting of PDEs (partial differential equations) plus ODEs (ordinary differential equations) into a much simpler problem where only ODEs have to be solved. This works fine as long as the whole plain is used as a domain for the diffusion equations, but we also want to be able to consider more general domains with, for example, areas where diffusion cannot take place (“holes”) or with boundaries. Also, Krottje (2003a) showed that in Hentschel and van Ooyen’s approach moving growth cones that secrete diffusible guidance molecules upon which they respond themselves cause the speed of growth to be strongly dependent on the diameter of the growth cone (a phenomenon that was called selfinteraction). The use of a quasisteadystate approximation will then result in heavily distorted dynamics.
Here, we present a general framework for the simulation of axon guidance together with novel numerical methods for carrying out the simulations. The two major ingredients of the modeling framework are the concentration fields of the guidance molecules and the finitedimensional state vectors representing the growth cones and target neurons. For the latter two, ODEs must be constructed that describe the interaction with the concentration fields. The dynamics of the fields is described by diffusion equations, where we allow for domains with holes or internal boundaries.
Numerical difficulties arise from small, moving sources for the diffusion equations (see Krottje, 2003a) and from the time integration of a system that is a combination of highly nonlinear, nonstiff ODEs and stiff diffusion equations (see Verwer and Sommeijer, 2001). To circumvent this last difficulty we consider the use of quasisteadystate approximations, and we will discuss some criteria on the validity of such approximations.
The organization of the paper is as follows. We start with a description of the simulation framework in Section 2. In Section 3 we will discuss some features of the underlying mathematical model and in Section 4 the numerical methods are discussed. Some simulation examples are given in Section 5. We will finish with a discussion in Section 6.
2. Simulation framework
2.1. States
We define states to be finitedimensional state vectors that represent objects that interact with the concentration fields of guidance molecules. These objects can be, for example, growth cones that move in response to the concentration fields, target neurons that act as sources of guidance molecules, or locations where artificial injection of guidance molecules takes place.

Sensitivity: Growth cones can respond to different guidance molecules. Their sensitivity to a particular molecule may vary over time (Shewan et al., 2002) and can be influenced by the concentration levels of other guidance molecules as well as by the level of signaling molecules inside the growth cone (Song and Poo, 1999).

Growth cone geometry: It is known that growth cones can change their size while moving through the environment (Rehder and Kater, 1996). The vector s could model how this process depends on the concentration fields, or it could model the way in which changes in growth cone size change the growth cone’s sensitivity or behavior.

Internal state of growth cone: Inside a growth cone biochemical reactions take place that determine the growth cone’s dynamics (Song et al., 1998; Song and Poo, 1999). With s, the concentrations of the different reactants and their effect on growth cone dynamics and axon guidance can be modeled.

Production rates: The rate at which target cells produce guidance molecules may depend on the concentration fields measured at the locations of the targets. The vector s can be used to describe such dependencies. Alternatively, s can describe production rates that are given explicitly as functions of time.
The functions G ^{ r }, G ^{ s } and G are used to model the different biological processes and mechanisms. We will now discuss the fields ρ_{ j } (j = 1, …, M).
2.2. Fields
A number of states is linked to a field. These states determine the total source function S _{tot}, which is the sum of source functions S _{ i }, each of them belonging to a single state (r, s)_{ i }. To further specify the form of the S _{ i }, we make use of a translation operator T _{y}, which can by applied to arbitrary functions η : Ω → ℝ and which is defined for y ∈ Ω by (T _{y}η)(x) = η(x − y) for all x ∈ Ω. For the source functions S _{ i } : Ω → ℝ, we make the assumption that S _{ i } = σ_{ i }(s _{ i })T _{ r i } S. Here, S is some general function profile and σ_{ i }(s _{ i }) ∈ ℝ denotes the production rate.
2.3. Coupling
We see that only the dynamics of state 1 depends on the state itself, which is reflected in having an ODE for its dynamics, while the dynamics of the other two states are given in a more explicit form.
3. Underlying mathematical model
In the functions G _{ i } we have to implement the different mechanisms that are involved in the behavior of the growth cones and targets when they measure the levels of particular concentration fields and their gradients. To complete the system we have to add initial conditions for the states u _{ i } and the fields ρ_{ j }.
3.1. Moving sources
Our framework also allows for the possibility that guidance molecules are released by the growth cones themselves, i.e., we allow for moving sources. Although the biological evidence for this is not so strong as for the release of guidance molecules by target cells, it is certainly not implausible. Growth cones secrete various chemicals that may operate as chemoattractants and chemorepellents. For example, migrating axons are capable of secreting neurotransmitters (Young and Poo, 1983), which have been implicated as chemoattractants (Zheng et al., 1994). The treatment of moving sources that respond to guidance molecules they themselves secrete is mathematically challenging and will be dealt with in the Appendix.
3.2. Quasisteadystate approximation
The original dynamical system, which had as its dependent variables the states u _{ i } and the fields ρ_{ j }, is now replaced by a dynamical system that has only the u _{ i } as its dependent variables. Although the system at hand is therefore reduced from an infinitedimensional to a finitedimensional system, evaluation of the righthand side still involves solving a infinitedimensional system. Determination of the values ρ_{ j }(r _{ i }) requires solving Eqs. (6’–7). From a numerical perspective the advantage is that we do not need a time integrator that can handle the combination of stiff PDEs and nonstiff ODEs, but we can simply make use of a standard explicit time integrator.
Some implications of using an approximation like (12) for (11) are discussed in Krottje (2003a). There the case of selfinteraction is considered, meaning that for a particular field a source is attached to a state and the dynamics of the state is determined by the same field. Here, we want to consider some more general criteria as to when such a quasisteadystate approximation might be valid for different parameter values of the diffusion rate d, the absorption rate k, and the speed a source moves through the domain v.
Hentschel and van Ooyen (1999) used the approximation on the basis of comparing the time scales of growth and diffusion. Here, however, the absorption parameter plays also a role. To determine criteria that take also k into account we will follow two approaches. In the first approach we consider the time needed for settingup a concentration field. In the second approach we compare the concentration profile produced by a point source moving with constant speed with its quasisteadystate approximation.
3.3. Field setup time
3.4. Field produced by moving source
Consider Eq. (11) with a point source that moves with constant speed v along the xaxis in positive direction, i.e., S(x, t) = δ(x − v t), with v = (ν, 0)^{T}. In the Appendix it is shown that we get a stable constant profile solution that moves also with constant speed v.
If we choose γ = 0.99, we get an indication of the radius r of the region around the source where the difference between the moving profile and the quasisteadystate solution is less than about 1%, given the values of the diffusion rate d, absorption rate k and moving speed v.
3.5. Typical parameter ranges
Parameter ranges.
Quantity  Symbol  Order of magnitude  Units 

Diffusion constant  d _{ j }  10^{−5} to 10^{−4}  mm^{2}/s 
Production rate  σ_{ ji }  10^{−7}  nMol/s 
Minimal concentration for gradient detection  ρ _{ min }  10^{−2} to 10^{−1}  nMol/l 
Maximal concentration for gradient detection  ρ_{ max }  100  nMol/l 
Minimal relative detectable gradient  L_{cone}V_{ ρj }/_{ρj }  0.01 to 0.02  
Growth cone diameter  L _{cone}  10^{−2} to 2 × 10^{−2}  mm 
Growth speed  v  10^{−6} to 10^{−4}  mm/s 
Growth range  L _{path}  10^{−1} to 1  mm 
Here, we used an expression for ρ_{ s } that is derived in the Appendix. We can derive a lower bound for the absorption constant k _{ j } by considering the ratio L _{cone}∂_{ r }ρ_{ s }(r)/ρ_{ s }(r) which decreases with r and increases with k _{ j }. If we assume it to be greater than 0.01, for all r ≤ 1, this yields a bound \(\sqrt {{k_j}/{d_j}} \ge 0.60\).
4. Numerical methods
In this section, we will consider the numerical methods we use for solving Eqs. (6–10). We will start with the spatial discretization for solving the field equations. This will be followed by a description of the time integration techniques.
For solving the field equations we use an unstructured spatial discretization based on an arbitrary set of nodes situated in the domain. This approach facilitates dealing with complex domains, refinement and adaptivity; the latter is needed in cases where we have moving sources with small support. A thorough description of the method can be found in Krottje (2003b); we will briefly outline it here.
4.1. Function approximation
Given function values on the nodes, we use a local leastsquares approximation technique to determine for every node a secondorder multinomial that is a local approximation of the function around that node. For this we use the function values on a number of neighboring nodes. Because every secondorder multinomial can be written as the linear combination of six basis functions, we must choose at least five neighbors for every node to determine such an approximating multinomial.
With this procedure a set of function values is mapped onto a set of local approximations around every node. If we assign to every node a part of the domain for which we assume the local approximation to be valid, such that the whole domain is covered, this results in a global approximation. For a given set of function values in a vector w ∈ ℝ^{ N }, we denote the global approximation by F(w) ∈ L _{1}(Ω), where L _{1}(Ω) is the space of integrable real functions defined on Ω ⊂ ℝ^{2}.
4.2. Voronoi diagrams
For choosing neighboring nodes of nodes, as well as for assigning parts of the domain to the nodes, we use the Voronoi diagram (Fortune, 1987). It assigns to every node a Voronoi cell, which is the set of points closer to the node than to every other node, hence dividing the domain and at the same time creating neighbors in a natural way.
Because a Voronoi diagram extends to all of ℝ^{2}, we will truncate it by connecting the nodes on the boundary by straight lines, resulting in a bounded diagram. From now on all our diagrams will be truncated ones, but we will still refer to them as Voronoi diagrams. Determination of such a diagram can be done in O(Nlog(N)) operations, where N is the number of nodes (de Berg et al., 2000). We store the diagram in a totally disconnected edge list (de Berg et al., 2000), so that searching neighboring nodes for every node becomes a process of O(N) operations.
4.3. Variational problem
A direct discretization of this problem is to minimize A(F(w), F(w)) − L(F(S), F(w)), over all w ∈ ℝ^{ N }, where S ∈ ℝ^{ N } is the vector of Svalues at the nodes. It can be shown (Krottje, 2003b) that sparse matrices ^A and ^L exist such that ½w ^{ T } ^A w = A(F(w), F(w)) and S ^{ T } ^L w = L(F(S), F(w)). If ^A is nonsingular the discrete problem has a unique solution w = ^A ^{−1} S. With the algorithm for finding the Voronoi diagram comes a lexicographical ordering of the nodes that will give the sparse matrices a band structure, which is advantageous when solving the system directly using an LUdecomposition.
Convergence tests show that the solution is secondorder convergent in the L ^{2}norm, with respect to the maximum distance between neighboring nodes (Krottje, 2003b).
4.4. Choosing nodes
To distribute nodes appropriately over a domain we make use of Lloyd’s algorithm (Du et al., 1999). This algorithm is based upon the determination of Voronoi diagrams and the process of shifting nodes to centroids of Voronoi cells. An alternating sequence of these two operations distributes the nodes equally over the domain, in the sense that distances between neighbors will tend to become equal throughout the diagram.
To achieve refinement at certain points, we use a variation of Lloyd’s algorithm. Here, after shifting the nodes to their centroids, an extra shift in the direction of neighboring nodes is added. To determine for a particular node which of its neighbors are attracting this node, all nodes are given an integer type. Nodes will then be attracted to the neighbors with higher type than their own type.
To get refinement around a certain point in the domain, a node is fixed at that point and several rings of decreasing node type are defined around it. The extended Lloyd’s algorithm then moves nodes around, which results in a refinement around the fixed node.
In contrast to methods where refinement is based on local error estimation, here refinement takes place around the source locations. This is done because we know in advance that only at those locations, and possibly at the boundary, refinement is required for optimal accuracy. Doing it in this way instead of using an error estimation process will speed up the refinement process.
Having discussed the spatial discretization method we will now focus on the time integration. We will consider three different cases that can be distinguished by the field dynamics in the model.
4.5. Time integration with static fields
We first consider the case with static fields only. In this case we only have ODEs which need the solutions of the fields for the evaluation of their righthand sides. These fields are determined at the start of the simulation by solving the elliptic equations, giving the approximations to the field solutions ρ_{1}, …, ρ_{ M }. After this the growth cone dynamics can be solved using a standard explicit ODE solver.
When combined with Eq. (19), evaluation of these field solutions and their gradients in the given r _{ i }, results in a closed algebraic system with respect to s _{ i }, ρ_{ j }(r _{ i }), ∂_{ x }ρ_{ j } (r _{ i }) and ∂_{ y }ρ_{ j } (r _{ i }). We will assume that this nonlinear system can be solved, although the solvability depends on the r _{ i } and the functions σ_{ ji }.
Therefore, to solve numerically the fields ρ_{ j } we first have to solve numerically the fields \(L_j^{  1}{T_{{{\rm{r}}_i}}}S\), using the spatial discretization above. After evaluation of these fields (i.e., their numerical approximations) and their derivatives in all locations r _{ i }, the algebraic system can be built by substituting (21) into (19). We can solve this system by using, for example, Newton iterations and use the s _{ i } to determine the solutions ρ_{ j }.
4.6. Quasisteadystate approximation
Here we use, as in the previous case, an explicit time integrator for the ODEs in (24). To evaluate the righthand side of the equations we need to solve the fields ρ_{ j } for given values of (r _{ i }, s _{ i })_{ n }, i = 1, …, N _{ o }, and t _{ n }, where n denotes the time level. To find these we have to determine the fields again by solving a nonlinear algebraic system as is done in the case with static fields. Here, the system will have as its unknowns the ρ_{ j }(r _{ i }), ∂_{ x }ρ_{ j }(r _{ i }) and ∂_{ y }ρ_{ j }(r _{ i }) for all combinations of fields ρ_{ j } and states r _{ i }, together with all s _{ i }, for i > N _{ o }.
In contrast to the case with static fields, every function evaluation in the righthand side of (24) requires solving Eq. (26) and evaluation of the resulting solution fields and their gradients. Also, because the source terms in (26) depend on the states u _{ i }, may be necessary to redefine the nodes used to solve the field equations. Therefore, solving such a system is computationally much more expensive than solving a system with static fields only.
4.7. Full system
Solving the full system, i.e., Eqs. (6–10), requires a numerical method that can deal with both the nonlinear, nonstiff ODEs and the stiff diffusion equations. Verwer and Sommeijer (2001) use for a system similar to the combination of (6) and (9) the RKC method, which is explicit and can deal with moderately stiff systems due to a long narrow stability region around the negative real axis. Lastdrager (2002) used a Rosenbrock method with approximate Jacobians for the same system so that effectively the field equations are integrated implicitly and the state equations explicitly, as with IMEX (IMplicitEXplicit) methods (Hundsdorfer and Verwer, 2003).
In the next section, we will show some example models. Although our framework can deal with nonstatic fields (as discussed earlier), in these examples we will consider only cases in which the fields are static.
5. Simulation examples
In this section, we will discuss simulations of some example models. We want to stress that the models used here are still simple and only serve to show the different possibilities of our framework. To model the growth cones and the sources of the guidance molecules, such as target cells, we have to choose state vectors (r _{ i }, s _{ i }) that characterize these objects and accompanying functions G that describe the dynamics through Eqs. (1) and (2).
5.1. Growth cone model
As a first example of a growth cone model we consider growth cones characterized by threedimensional state vectors. To the position r _{ i } = (x, y) we add a variable representing the orientation angle s _{ i } = ϕ ∈ [0, 2π) of the growth cone. This gives our model growth cone a growth direction, which it has to adjust in order to steer. It gives the opportunity to build in some kind of ‘stiffness’, the inability to undergo instant changes in growth direction.
5.2. Field sources
In the examples we will assume that the fields are produced by sources that are not moving and not changing their behavior in time. Therefore, it will serve to include in their state vectors only their positions r _{ i } ∈ ℝ^{2} and keep them constant in time r _{ i } = G _{ i } ^{ r } (t) ≡ r _{ i } ^{0} .
5.3. Example 1. Axon guidance in a simple concentration field
We will now consider an example simulation, with a single concentration field and a single growth cone. For the domain Ω we take the unit circle and put a source at (0.5, 0). This source produces at a production rate σ_{11} = 1.0 × 10^{−4} a field ρ_{1} with diffusion coefficient d _{1} = 1.0 × 10^{−4} and absorption parameter k _{1} = 1.0 × 10^{−4}. For the width of the source we take w = 0.02.
The growth cone is modeled by using system (29) with the functions λ_{1} set to λ_{1} ≡ 1, which means that ϕ_{ g } = arg∇ρ_{1}. Further we use the parameter values v = 1.0 × 10^{−5} and ℓ= 0.02.
If we compare this result with pictures of similar experiments with real axon growth (Dodd and Jessell, 1988), we see that real axons often start to grow away from the initial area before they seem to react to the attracting field. This could mean that real axons have a higher stiffness than the stiffness we used in the left panel of Fig. 5. We therefore increased the stiffness by setting ℓ = 0.1. This results in the paths shown in the middle panel of Fig. 5. While this gives a somewhat better result, it seems not realistic to increase the stiffness this far, because growth cones are capable of making sharper turns (turning angles up to 90°; Song et al., 1998, Ming et al., 2001) than those seen in the middle panel of Fig. 5.
5.4. Example 2. Axon guidance in a complex concentration field
We will now consider a variation of the previous example where the domain has been changed from a simple circular domain to a more complex domain with four holes in it. These holes might represent blood vessels or cells where the axons have to grow around and that are impenetrable to the diffusive guidance molecules.
Although the axons grow nicely around the holes, there is actually no mechanical force in the model that prevents the growth cones from entering the holes. Here the growth cone dynamics alone was sufficient to keep the growth cones outside the holes. However, if is bigger, the growth cones will need more space to turn, and might enter the holes if not stopped by a hard boundary.
5.5. Example 3. Axon guidance with internal growth cone dynamics
In this example we will extend our cone dynamics by adding another variable. In the previous examples the ideal direction, based on the sensed gradients, is directly translated into a change of direction. In real growth cones, however, signaling pathways inside the growth cone are responsible for this translation. We now incorporate such signaling pathways and represent it by a single variable α ∈ [−1, 1], where α < 0 means steering to the left and α > 0 steering to the right. The growth cone translates the ideal direction into the signaling pathway dynamics in a way that is similar to the way that the ideal direction is translated into the direction dynamics in the previous examples.
5.6. Example 4. Axon guidance with membranebound guidance molecules in topographic map formation
In our last example we consider a more complicated model of a phenomenon that is called topographic mapping (van Ooyen, 2003). Many neuronal connections are made so as to form a topographic map of one structure onto another. That is, neighboring cells in one structure make connections to neighboring cells in the other structure. An example of such a map is the direct projection of the retina onto the optic tectum in the brain of nonmammalian vertebrates (Gaze, 1958). One explanation for the formation of topographic maps that has received strong experimental support is that it is based on the matching of gradients of receptors and their ligands (O’Leary and Wilkinson, 1999; Wilkinson, 2001). For the retinotectal projection, there is a gradient across the retina in the number of Eph receptors on the growth cones of the retinal neurons. A similar but opposite gradient is found across the tectum in the number of membranebound ephrin molecules (the ligands for Eph receptors) on the tectal neurons. Axons grow out so that growth cones with a low number of receptors come to connect to tectal cells with a high number of ligand molecules, and vice versa.
Here the function Sgm_{ n } is defined by Sgm_{ n }(x) = x ^{ n }/(1 + x ^{ n }). Finally, we will assume that the growth is completely inhibited if both c _{ x } < 0.8 and c _{ y } < 0.8.
To visualize the conservation of spatial order we have color coded the initial locations and end locations of the paths, i.e., begin and end points of a path have the same color. The result of this is displayed in the left panel and it clearly shows that the ‘A’ is transferred from the initial area (at the left) to the final area (at the right).
The combination of the membranebound ligand fields ρ_{4} and ρ_{5} with the receptor densities β_{ x } and β_{ y } determines what the topographic mapping will look like. Using a model like this for exploring different possibilities for the concentration fields can give us more insight into the forms of the fields and mechanisms involved in topographic map formation.
6. Discussion
In this paper, we have presented a framework for modeling axon guidance. Unlike for the study of electrical activity in neurons and neuronal networks, such a general framework did not exist. Our framework allows for the relatively straightforward and fast modeling and simulation of axon guidance and its underlying mechanisms. For example, mechanisms that ‘translate’ concentration levels of guidance molecules (or gradients thereof), measured at the location of the growth cone, into growth speed, growth direction and sensitivity for particular concentration fields can easily be incorporated. A major challenge in the study of axon guidance is to understand quantitatively how the many molecules and mechanisms involved in axon guidance act in concert to generate complex patterns of neuronal connections. The framework we have developed contributes to this challenge by providing a general simulation tool in which a wide range of models can be implemented and explored.
Our framework has three basic ingredients: the domain, the concentration fields and the states. The domain models the physical environment where the neurons, axons, and fields live in; the domain can have a complicated geometry with piecewise smooth boundaries and holes. The fields are defined on the domain and represent the time varying concentration fields of guidance molecules that are subject to diffusion and absorption. The states model the growth cones and targets cells and consist of finitedimensional vectors for which the dynamics are given in the form of ODEs that model the mechanisms involved in axon guidance.
Specific numerical methods have been developed for solving the systems of equations that typically arise in models of axon guidance. With respect to time integration of the full system, a method is needed that can handle the combination of stiff diffusion equations (describing the concentration fields) and nonstiff, nonlinear differential equations (describing the states). For this a secondorder RungeKutta IMEX scheme is used. In case of static fields or a quasisteadystate approximation an explicit time integrator will suffice, for which we use the classical fourthorder RungeKutta method.
The spatial discretizations required for solving the elliptic field equations that arise after discretization in time are based on arbitrary node sets. Voronoi diagrams are used for the selection of suitable node sets as well as for the discretization of the equations. To speed up the node selection process, refinement and adaptivity of the discretization are based only upon the location of the highly localized sources.
We have implemented the framework and the numerical algorithms in a set of Matlab programs. In these programs one can simulate a wide range of models by defining appropriate Matlab datastructures and solve them by applying the spatial and temporal numerical solvers. At the moment, the code is typical research code without extensive documentation, but we aim to create a more userfriendly version that can be made available to the wider research community.
Possible extensions of our framework include the incorporation of randomness in the guidance of the axons and the possibility that boundaries (of impenetrable holes, for example) can produce guidance molecules. The latter extension would make it possible to model also tissues, rather than individual cells, that attract or repel axons.
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